Mean Convex Neighborhood Conjecture

• Mean Convex Neighborhood Conjecture is a geometric property stating that near cylindrical singularities in mean curvature flow, a full space–time region maintains mean convexity.
• It employs asymptotic, spectral, and barrier methods to classify ancient, low-entropy flows and to ensure a continuous, mean convex level-set structure near singular points.
• The conjecture’s implications include guaranteeing uniqueness of flow continuation and precluding fattening, thereby providing a robust framework for analyzing singularities in all dimensions.

The Mean Convex Neighborhood Conjecture addresses the structure of singularities formed by the mean curvature flow (MCF) of hypersurfaces, asserting that near any generic singularity—where the tangent flow is a multiplicity-one shrinking cylinder—a full space–time neighborhood exists in which the flow is mean convex. This property provides critical control over the evolution near singular points, ensuring local geometric rigidity, uniqueness of continuation, and precluding pathological phenomena such as “fattening” of the level-set flow. Rigorous confirmation of the conjecture has been achieved in all dimensions and for all cylindrical singularities, supported by classification results for ancient low-entropy flows and detailed analysis of the structure of Brakke flows in the presence of cylindrical singularities.

1. Formal Statement and Geometric Setting

The conjecture is formulated for unit-regular, integral Brakke flows in $\mathbb{R}^{n+1}$, focusing on singular points $(p_0, t_0)$ where the tangent flow is a multiplicity-one round shrinking cylinder. For such a point, the conjecture asserts the existence of an open neighborhood $U\subset\mathbb{R}^{n+1}$ and $\varepsilon>0$ such that:

• Each time-slice $(\mathrm{spt}\, M)_t \cap U$ is the level set $\{p \in U : u(p) = t\}$ for some continuous $u: U \rightarrow \mathbb{R}$,
• On regular points, the mean curvature vector satisfies $\mathbf{H}(p,t) = \nabla u(p)/|\nabla u(p)|^2$ with $H \ge 0$,
• All tangent flows at points in $U \times (t_0 - \varepsilon, t_0 + \varepsilon)$ are multiplicity-one round cylinders $\mathbb{R}^{k'} \times S^{n-k'}$ for $k' \le k$ (Bamler et al., 31 Dec 2025, Choi et al., 2018, Gang, 2018).

This structure theorem situates mean convexity as a stable, robust local property around cylindrical singularities and provides a framework for classifying and understanding singular behavior in MCF.

2. Preliminaries: Weak Solutions and Ancient Flows

The mean curvature flow is often analyzed in the framework of Brakke flows, which generalize classical smooth solutions to accommodate singularities and weak motions. An $n$-dimensional Brakke flow $\{\mu_t\}$ in $\mathbb{R}^{n+1}$ is a family of Radon measures satisfying a monotonicity inequality involving the mean curvature vector and test functions. Integral, unit-regular Brakke flows are required: any point of Gaussian density 1 is smooth, and the structure avoids higher-multiplicity or non-orientable phenomena (Choi et al., 2018, Bamler et al., 31 Dec 2025).

A cornerstone of the analysis is the classification of ancient low-entropy flows—eternal Brakke flows that can arise as blow-up limits near singularities. In $\mathbb{R}^3$, any ancient, unit-regular, cyclic, integral Brakke flow with entropy at most $\sqrt{2\pi/e}+\delta$ is a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval (Choi et al., 2018). In higher dimensions, ancient cylindrical solutions include ovals, the bowl soliton, and new families such as flying wing solitons (Bamler et al., 31 Dec 2025).

3. Proof Strategies and Key Structural Theorems

The mean convex neighborhood property is proved by a combination of asymptotic, spectral, and barrier arguments, often in the rescaled (parabolically blown-up) setting. The main structural theorem establishes that near any singularity with a cylindrical tangent flow, the Brakke flow admits a continuous level-set function whose regular slices are mean convex, and all nearby tangent flows remain cylindrical (Bamler et al., 31 Dec 2025).

The proof proceeds by:

• A refined asymptotic analysis of ancient cylindrical flows, showing that all such flows are convex, non-collapsed, and rotationally symmetric;
• The “leading mode condition,” employing a sharp control of modes in the spectral decomposition of the linearized operator (Ornstein–Uhlenbeck type) on the cylinder, to classify local geometry near the singularity;
• An induction-over-thresholds scheme, incrementally propagating local geometric control inward by maximum principle and comparison arguments;
• Establishment of a canonical neighborhood theorem: every regular point near the singularity has a neighborhood closely modeled on one of the classified ancient flows, with quantitative estimates controlling structure in both space and time (Bamler et al., 31 Dec 2025).

In the non-degenerate case, rescaling and spectral analysis reveal that the profile function of the MCF converges to the model cylinder, ensuring strict mean convexity in a fixed region up to the singular time. In degenerate (even-mode) cases, a sign-preserving region $v > 1$ persists, again guaranteeing mean convexity. Fully degenerate cases are conjectured to produce simultaneous blow-up across a small ball, with inductive evidence paralleling known results in semilinear heat flow (Zhou, 2021).

4. Canonical Neighborhoods and Uniqueness

The canonical neighborhood theorem asserts that, near any multiplicity-one cylindrical singularity, each point of the flow is locally modeled by some ancient, asymptotically cylindrical solution (oval, bowl soliton, or flying wing), and the flow is mean-convex in this region (Bamler et al., 31 Dec 2025). As a consequence, the singularity is isolated in space–time.

This structural control directly underpins several uniqueness results:

• If all singularities in a level-set flow are cylindrical/spherical, local mean convexity excludes non-uniqueness phenomena (“fattening”) (Choi et al., 2018, White, 2011).
• For embedded two-spheres, assuming multiplicity-one at singularities, all tangent flows must be spheres or cylinders, establishing well-posedness for MCF through singularities (Choi et al., 2018).
• In dimensions $n \geq 2$, the singular set has parabolic Hausdorff dimension at most $n-2$, and near each singularity, the mean curvature is nearly convex, in the quantitative sense that the smallest principal curvature $\kappa_1$ satisfies $\kappa_1 \geq -\epsilon H - C$ for arbitrarily small $\epsilon > 0$ (White, 2011).

5. Broader Geometric and Analytical Context

The mean convex neighborhood property links to geometric rigidity and collapse phenomena beyond the immediate setting of MCF. In the context of Riemannian geometry, Gromov formulated related conjectures asserting that domains with strictly positive mean curvature boundary admit neighborhoods that can be continuously retracted to skeleta of codimension at least two, with the move-distance bounded by $1/H_0$ (Gromov, 2018). These conjectures reflect an interplay between lower mean curvature bounds and topological constraints, analogously to Myers-type theorems or Uryson-width bounds in scalar curvature geometry.

Motivating the conjecture are sharp comparison results, e.g., the in-ball inequality, extrinsic extremality of spheres, and filling-radius bounds, exhibiting the collapse of such domains to lower-dimensional sets under mean-convexity assumptions. The MCF results thus resonate with broader themes in geometric analysis, emphasizing the rigidity imparted by global curvature conditions.

6. Significance and Further Directions

Resolution of the Mean Convex Neighborhood Conjecture has led to a comprehensive classification of singularities in MCF, directly informing uniqueness, stability, and surgery constructions in geometric flows. The classification of ancient, asymptotically cylindrical flows provides canonical local models, anchoring future work in the analysis of generalized flows, high-codimension settings, and boundary value problems.

Open directions include the pursuit of sharp constants in Gromov’s retraction conjectures, regularity of collapsing maps, extension to non-spin manifolds, higher-codimension analogues, and a full proof in certain degenerate regimes of MCF where simultaneous blowup is conjectured but not fully established (Zhou, 2021, Gromov, 2018). Nonetheless, the mean convex neighborhood property remains a pivotal result in the structural analysis of geometric evolution equations, providing both a local and global organizing principle for singularity theory in mean curvature flow.